The odd chromatic number of a planar graph is at most 8

Petr, Jan; Portier, Julien

doi:10.48550/arxiv.2201.12381

Cited by 9 publications

(13 citation statements)

References 2 publications

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“…The minimum number of colors in any odd coloring of G is the odd chromatic number of G, denoted χ o (G). This new graph parameter spurred instant interest among graph theorists (see [11,6,29,12,10,15]). Clearly χ(G) ≤ χ o (G) ≤ χ pcf (G), where the latter inequality comes from the obvious fact that every proper conflict-free coloring is odd.…”

Section: Introductionmentioning

confidence: 99%

Remarks on proper conflict-free colorings of graphs

Caro¹,

Petruševski²,

Škrekovski³

2022

Preprint

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A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χ pcf (G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χ pcf (G) in terms of other graph parameters. In particular, we show that χ pcf (G) ≤ 5∆(G) 2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4 ≤ k ≤ 6. The paper concludes with few open problems.

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Section: Introductionmentioning

confidence: 99%

Remarks on proper conflict-free colorings of graphs

Caro¹,

Petruševski²,

Škrekovski³

2022

Preprint

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“…It is worth noting that their proof of this key step relies on Theorem 4 in Aashtab, Akbari, Ghanbari, and Shidani [1], which itself relies on the Four-Color Theorem [2,12]. Building on work of Caro, Petruševski, and Škrekovski [3], Petr and Portier [10] further proved that 8 colors suffice for all planar graphs; that is, every planar graph admits an odd 8-coloring.…”

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confidence: 99%

A Note on Odd Colorings of 1-Planar Graphs

Cranston¹,

Lafferty²,

Song³

2022

Preprint

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A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petruševski and Škrekovski, who proved that every planar graph admits an odd 9-coloring; they also conjectured that every planar graph admits an odd 5-coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petruševski, and Škrekovski. Building on the work of Caro, Petruševski, and Škrekovski, Petr and Portier then further proved that every planar graph admits an odd 8-coloring. In this note we prove that every 1-planar graph admits an odd 23-coloring, where a graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge.

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“…First, Cranston [8] established several results for sparse graphs, and as a corollary, he obtained bounds 6 and 5 for planar graphs of girth 6 and at least 7, respectively. Then, Caro et al [4] established the bound 8 for planar graphs with specific properties, and finally, Petr and Portier [19] proved the bound 8 for all planar graphs.…”

Section: Discussionmentioning

confidence: 99%

Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods

Fabrici¹,

Lužar²,

Rindošová³

et al. 2022

Preprint

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A conflict-free coloring of a graph with respect to open (resp., closed) neighborhood is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed) neighborhood. Similarly, a unique-maximum coloring of a graph with respect to open (resp., closed) neighborhood is a coloring of vertices such that for every vertex the maximum color appearing in its open (resp., closed) neighborhood appears exactly once. There is a vast amount of literature on both notions where the colorings need not be proper, i.e., adjacent vertices are allowed to have the same color.In this paper, we initiate a study of both colorings in the proper settings with the focus given mainly to planar graphs. We establish upper bounds for the number of colors in the class of planar graphs for all considered colorings and provide constructions of planar graphs attaining relatively high values of the corresponding chromatic numbers. As a main result, we prove that every planar graph admits a proper unique-maximum coloring with respect to open neighborhood with at most 10 colors, and give examples of planar graphs needing at least 6 colors for such a coloring. We also establish tight upper bounds for outerplanar graphs. Finally, we provide several new bounds also for the improper setting of considered colorings.

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The odd chromatic number of a planar graph is at most 8

Cited by 9 publications

References 2 publications

Remarks on proper conflict-free colorings of graphs

Remarks on proper conflict-free colorings of graphs

A Note on Odd Colorings of 1-Planar Graphs

Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods

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